reserve A for preIfWhileAlgebra,
  C,I,J for Element of A;
reserve S for non empty set,
  T for Subset of S,
  s for Element of S;

theorem Th81:
  pr1(S, the carrier of A) is complying_with_empty-instruction &
  pr1(S, the carrier of A) is complying_with_catenation &
  pr1(S, the carrier of A) complies_with_if_wrt T &
  pr1(S, the carrier of A) complies_with_while_wrt T
proof
  set f = pr1(S, the carrier of A);
  thus
  for s being Element of S holds f.(s, EmptyIns A) = s by FUNCT_3:def 4;
  hereby
    let s be Element of S;
    let I1,I2 be Element of A;
    thus f.(s,I1 \; I2) = s by FUNCT_3:def 4
      .= f.(s,I1) by FUNCT_3:def 4
      .= f.(f.(s,I1),I2) by FUNCT_3:def 4;
  end;
  hereby
    let s be Element of S;
    let C, I1,I2 be Element of A;
    f.(s,if-then-else(C,I1,I2)) = s by FUNCT_3:def 4
      .= f.(s,C) by FUNCT_3:def 4;
    hence
    (f.(s,C) in T implies f.(s,if-then-else(C,I1,I2)) = f.(f.(s,C),I1))
    & (f.(s,C) nin T implies f.(s,if-then-else(C,I1,I2)) = f.(f.(s,C),I2))
    by FUNCT_3:def 4;
  end;
  let s be Element of S;
  let C, I be Element of A;
  f.(s,C) = s by FUNCT_3:def 4;
  hence thesis by FUNCT_3:def 4;
end;
